Count data are often used in recommender systems: they are widespread (song play counts, product purchases, clicks on web pages) and can reveal user preference without any explicit rating from the user. Such data are known to be sparse, over-dispersed and bursty, which makes their direct use in recommender systems challenging, often leading to pre-processing steps such as binarization. The aim of this paper is to build recommender systems from these raw data, by means of the recently proposed compound Poisson Factorization (cPF). The paper contributions are three-fold: we present a unified framework for discrete data (dcPF), leading to an adaptive and scalable algorithm; we show that our framework achieves a trade-off between Poisson Factorization (PF) applied to raw and binarized data; we study four specific instances that are relevant to recommendation and exhibit new links with combinatorics. Experiments with three different datasets show that dcPF is able to effectively adjust to over-dispersion, leading to better recommendation scores when compared with PF on either raw or binarized data.